Subgroups — C₂²×#557 #62709

Order 256.

3707 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#557	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₄⋊Q₈	C₂	1	3	yes	maximal
3	C₂²×#125	C₂	1	6	yes	maximal
4	C₂²×C₄×Q₈	C₂	1	3	yes	maximal
5	C₂²×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄×Q₈	C₂	1	2	yes	maximal
7	C₂³×Q₈	C₂²	1	3	yes	—
8	C₂³×Q₈	C₂²	1	1	yes	characteristic
9	#557	C₂²	1	256	yes	direct product factor
10	C₂×C₄⋊Q₈	C₂²	1	72	yes	—
11	C₂×#125	C₂²	1	144	yes	—
12	C₂×C₄×Q₈	C₂²	1	48	yes	—
13	C₂×C₄×Q₈	C₂²	1	72	yes	—
14	C₂×C₄×Q₈	C₂²	1	24	yes	—
15	C₂²×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄	C₂²	1	3	yes	—
17	C₂²×C₄⋊C₄	C₂²	1	6	yes	—
18	C₂²×C₄⋊C₄	C₂²	1	6	yes	—
19	C₂²×C₄⋊C₄	C₂²	1	6	yes	—
20	C₂²×C₄²	C₂²	1	6	yes	—
21	C₂²×C₄²	C₂²	1	3	yes	—
22	C₂²×Q₈	—	2	24	no	—
23	C₂²×Q₈	Q₈	1	16	yes	complemented
24	C₂²×Q₈	C₂³	1	36	yes	—
25	C₂²×Q₈	C₂³	1	12	yes	—
26	C₂³×C₄	C₂³	1	3	yes	—
27	C₂³×C₄	C₂³	1	6	yes	—
28	C₂³×C₄	C₂³	1	2	yes	—
29	C₂³×C₄	C₂³	1	3	yes	—
30	C₂³×C₄	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈	C₂³	1	192	yes	—
32	#125	C₂³	1	384	yes	—
33	C₄×Q₈	C₂³	1	128	yes	—
34	C₄×Q₈	C₂³	1	192	yes	—
35	C₄×Q₈	C₂³	1	64	yes	—
36	C₂×C₄⋊C₄	C₂³	1	12	yes	—
37	C₂×C₄⋊C₄	C₂³	1	72	yes	—
38	C₂×C₄⋊C₄	C₂³	1	36	yes	—
39	C₂×C₄⋊C₄	C₂³	1	72	yes	—
40	C₂×C₄⋊C₄	C₂³	1	72	yes	—
41	C₂×C₄²	C₂³	1	72	yes	—
42	C₂×C₄²	C₂³	1	36	yes	—
43	C₂⁴	C₂⁴	1	1	yes	characteristic
44	C₂×Q₈	—	2	144	no	—
45	C₂×Q₈	C₂×Q₈	1	96	yes	complemented
46	C₂×Q₈	C₂⁴	1	48	yes	—
47	C₂×Q₈	C₂⁴	1	16	yes	—
48	C₂²×C₄	—	2	24	no	—
49	C₂²×C₄	—	2	12	no	—
50	C₂²×C₄	—	2	8	no	—
51	C₂²×C₄	C₂×Q₈	1	24	yes	—
52	C₂²×C₄	#41	1	8	yes	—
53	C₂²×C₄	C₂⁴	1	12	yes	—
54	C₂²×C₄	C₂⁴	1	18	yes	—
55	C₂²×C₄	C₂⁴	1	36	yes	—
56	C₂²×C₄	C₂⁴	1	18	yes	—
57	C₂²×C₄	C₂⁴	1	6	yes	—
58	C₄⋊C₄	C₂⁴	1	16	yes	—
59	C₄⋊C₄	C₂⁴	1	96	yes	—
60	C₄⋊C₄	C₂⁴	1	48	yes	—
61	C₄⋊C₄	C₂⁴	1	96	yes	—
62	C₄⋊C₄	C₂⁴	1	96	yes	—
63	C₄²	C₂⁴	1	96	yes	—
64	C₄²	C₂⁴	1	48	yes	—
65	C₂³	C₂²×Q₈	1	4	yes	—
66	C₂³	C₂×#41	1	4	yes	—
67	C₂³	2₋¹⁺⁴	1	4	yes	—
68	C₂³	C₂⁵	1	3	yes	—
69	Q₈	—	2	96	no	—
70	Q₈	C₂²×Q₈	1	64	yes	complemented
71	C₂×C₄	—	2	72	no	—
72	C₂×C₄	—	2	24	no	—
73	C₂×C₄	—	2	36	no	—
74	C₂×C₄	C₂²×Q₈	1	72	yes	—
75	C₂×C₄	C₂×#41	1	24	yes	—
76	C₂×C₄	C₂⁵	1	8	yes	—
77	C₂×C₄	C₂⁵	1	12	yes	—
78	C₂×C₄	C₂⁵	1	24	yes	—
79	C₂×C₄	C₂⁵	1	12	yes	—
80	C₂×C₄	C₂⁵	1	4	yes	—
81	C₂²	#557	1	16	yes	direct product factor
82	C₂²	C₂³×Q₈	1	6	yes	—
83	C₂²	C₂²×#41	1	6	yes	—
84	C₂²	C₂×2₋¹⁺⁴	1	6	yes	—
85	C₂²	C₂⁶	1	1	yes	characteristic
86	C₄	—	2	24	no	—
87	C₄	—	2	8	no	—
88	C₄	—	2	12	no	—
89	C₄	C₂³×Q₈	1	24	yes	—
90	C₄	C₂²×#41	1	8	yes	—
91	C₂	C₂×#557	1	12	yes	direct product factor
92	C₂	C₂⁴×Q₈	1	1	yes	characteristic
93	C₂	C₂³×#41	1	1	yes	characteristic
94	C₂	C₂²×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#557	C₂	direct (N × H)
9. #557	C₂²	direct (N × H)
23. C₂²×Q₈	Q₈	semidirect
45. C₂×Q₈	C₂×Q₈	semidirect
70. Q₈	C₂²×Q₈	semidirect
81. C₂²	#557	direct (N × H)
91. C₂	C₂×#557	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#557	2
C₂²×C₄⋊Q₈	2
C₂²×#125	2
C₂²×C₄×Q₈	2

Subgroups — C22×#557 #62709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#557 #62709